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A334079 Total area of all triangles such that p + q = 2*n, p < q (p, q prime), with base (q - p) and height p. 1
0, 0, 0, 3, 6, 5, 12, 30, 34, 42, 54, 81, 72, 78, 126, 78, 168, 224, 84, 201, 332, 279, 252, 571, 462, 339, 760, 441, 522, 1002, 312, 915, 1194, 282, 1116, 1410, 810, 1233, 1778, 1092, 1206, 2332, 1218, 1212, 3396, 1536, 1404, 3017, 1572, 2250, 3988, 2055 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,4

LINKS

Table of n, a(n) for n=1..52.

Eric Weisstein's World of Mathematics, Goldbach Partition

Wikipedia, Goldbach's conjecture

Index entries for sequences related to Goldbach conjecture

Index entries for sequences related to partitions

FORMULA

a(n) = Sum_{i=1..n-1} i * (n-i) * c(i) * c(2*n-i), where c is the prime characteristic (A010051).

EXAMPLE

a(4) = 3; 2*4 = 8 has one Goldbach partition into distinct parts (5,3). The area of the triangle is then (5 - 3)*3/2 = 3.

a(8) = 30; 2*8 = 16 has two Goldbach partitions into distinct parts, (13,3) and (11,5). The sum of the areas of the two triangles is (13 - 3)*3/2 + (11 - 5)*5/2 = 15 + 15 = 30.

MATHEMATICA

Table[Sum[i (n - i) (PrimePi[i] - PrimePi[i - 1]) (PrimePi[2 n - i] - PrimePi[2 n - i - 1]), {i, n - 1}], {n, 60}]

CROSSREFS

Cf. A010051.

Sequence in context: A048724 A292682 A334748 * A328403 A199126 A247569

Adjacent sequences:  A334076 A334077 A334078 * A334080 A334081 A334082

KEYWORD

nonn,easy

AUTHOR

Wesley Ivan Hurt, Apr 13 2020

STATUS

approved

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Last modified April 19 17:46 EDT 2021. Contains 343117 sequences. (Running on oeis4.)